Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Stochastic Radon operators in porous media hydrodynamics

Authors: George Christakos and Dionissios T. Hristopulos
Journal: Quart. Appl. Math. 55 (1997), 89-112
MSC: Primary 76S05; Secondary 60G60, 76M35, 86A05
DOI: https://doi.org/10.1090/qam/1433754
MathSciNet review: MR1433754
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Abstract: A space transformation approach is established to study partial differential equations with space-dependent coefficients modelling porous media hydrodynamics. The approach reduces the original multi-dimensional problem to the one-dimensional space and is developed on the basis of Radon and Hilbert operators and generalized functions. In particular, the approach involves a generalized spectral decomposition that allows the derivation of space transformations of random field products. A Plancherel representation highlights the fact that the space transformation of the product of random fields inherently contains integration over a ``dummy'' hyperplane. Space transformation is first examined by means of a test problem, where the results are compared with the exact solutions obtained by a standard partial differential equation method. Then, exact solutions for the flow head potential in a heterogeneous porous medium are derived. The stochastic partial differential equation describing three-dimensional porous media hydrodynamics is reduced into a one-dimensional integro-differential equation involving the generalized space transformation of the head potential. Under certain conditions the latter can be further simplified to yield a first-order ordinary differential equation. Space transformation solutions for the head potential are compared with local solutions in the neighborhood of an expansion point which are derived by using finite-order Taylor series expansions of the hydraulic log-conductivity.

References [Enhancements On Off] (What's this?)

  • [1] I. Bialynicki-Birula, M. Cieplak, and J. Kaminski, Theory of Quanta, Oxford University Press, New York, N. Y., 1992
  • [2] G. Christakos, The space transformations and their applications in systems modelling and simulation, Proc. 12th Intern. Confer, on Modelling and Simulation (AMSE) 1 (3), Athens, Greece, 1984, pp. 49-68
  • [3] G. Christakos, Random Field Models in Earth Sciences, Academic Press, San Diego, CA, 1992
  • [4] G. Christakos and D. T. Hristopulos, Stochastic space transformation techniques in subsurface hydrology-Part 2: Generalized spectral decompositions and Plancherel representations, Stochastic Hydrology and Hydraulics 8, no. 2, 117-138 (1994)
  • [5] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • [6] I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Translated by Eugene Saletan, Academic Press, New York-London, 1964. MR 0166596
  • [7] Sigurdur Helgason, The Radon transform, Progress in Mathematics, vol. 5, Birkhäuser, Boston, Mass., 1980. MR 573446
  • [8] F. John, Plane Waves and Spherical Means, Springer-Verlag, New York, N. Y., 1955
  • [9] Michel Loève, Probability theory, Third edition, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. MR 0203748
  • [10] Adrian E. Scheidegger, The physics of flow through porous media, Revised edition, The Macmillan Co., New York, 1960. MR 0127717
  • [11] A. K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, Englewood Cliffs, NJ, 1989

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DOI: https://doi.org/10.1090/qam/1433754
Article copyright: © Copyright 1997 American Mathematical Society

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